Blog
About

130
views
0
recommends
+1 Recommend
0 collections
    8
    shares
      • Record: found
      • Abstract: found
      • Article: found
      Is Open Access

      Very Large-Scale Singular Value Decomposition Using Tensor Train Networks

      Preprint

      Read this article at

      Bookmark
          There is no author summary for this article yet. Authors can add summaries to their articles on ScienceOpen to make them more accessible to a non-specialist audience.

          Abstract

          We propose new algorithms for singular value decomposition (SVD) of very large-scale matrices based on a low-rank tensor approximation technique called the tensor train (TT) format. The proposed algorithms can compute several dominant singular values and corresponding singular vectors for large-scale structured matrices given in a TT format. The computational complexity of the proposed methods scales logarithmically with the matrix size under the assumption that both the matrix and the singular vectors admit low-rank TT decompositions. The proposed methods, which are called the alternating least squares for SVD (ALS-SVD) and modified alternating least squares for SVD (MALS-SVD), compute the left and right singular vectors approximately through block TT decompositions. The very large-scale optimization problem is reduced to sequential small-scale optimization problems, and each core tensor of the block TT decompositions can be updated by applying any standard optimization methods. The optimal ranks of the block TT decompositions are determined adaptively during iteration process, so that we can achieve high approximation accuracy. Extensive numerical simulations are conducted for several types of TT-structured matrices such as Hilbert matrix, Toeplitz matrix, random matrix with prescribed singular values, and tridiagonal matrix. The simulation results demonstrate the effectiveness of the proposed methods compared with standard SVD algorithms and TT-based algorithms developed for symmetric eigenvalue decomposition.

          Related collections

          Author and article information

          Journal
          2014-10-25
          2014-12-13
          1410.6895 10.1137/140983410

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

          Custom metadata
          15A18, 65F15, 65F30
          SIAM. J. Matrix Anal. Appl. 36(3):994-1014, 2015
          math.NA cs.NA

          Numerical & Computational mathematics

          Comments

          Comment on this article